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Let \mathfrak {g} be a complex simple Lie algebra. A simple \mathfrak {g}-module is called minimal if the associated variety of its annihilator ideal coincides with the closure of the minimal nilpotent coadjoint orbit. The main result of this paper is a classification of minimal highest weight modules for \mathfrak {g}. This classification extends the work of Joseph [Ann. Sci. École Norm. Sup. (4) 31 (1998), 17–45], which focused on categorizing minimal highest weight modules annihilated by completely prime ideals. Furthermore, we have determined the associated varieties of these modules. In other words, we have identified all possible weak quantizations of minimal orbital varieties. [ABSTRACT FROM AUTHOR]

The Gelfand–Kirillov dimension is a well established quantity to classify the growth of infinite dimensional algebras. In this article we introduce the algebraic entropy for path algebras. For the path algebras, Leavitt path algebras and the path algebra of the extended (double) graph, we compare the Gelfand–Kirillov dimension and the entropy. We show that path algebras over finite graphs can be classified to be of finite dimension, finite Gelfand–Kirillov dimension or finite algebraic entropy. We show indeed how these three quantities are dependent on cycles inside the graph. Moreover we show that the algebraic entropy is conserved under Morita equivalence but perhaps for a different filtration. In addition we give several examples of the entropy in path algebras and Leavitt path algebras. [ABSTRACT FROM AUTHOR]

Let g be a classical complex simple Lie algebra and q be a parabolic subalgebra. Let M be a generalized Verma module induced from a one dimensional representation of q . Such M is called a scalar generalized Verma module. In this paper, we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand–Kirillov dimension of the corresponding highest weight modules. [ABSTRACT FROM AUTHOR]

In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of G = GLn(ℂ) possessing the minimal Gelfand–Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of G of type (n − 1,1). We give the transition matrix between the two bases for the corresponding coherent families. [ABSTRACT FROM AUTHOR]

Let p be a prime number and F a totally real number field unramified at places above p. Let r ‾ : Gal (F ‾ / F) → GL 2 ( F p ‾) be a modular Galois representation which satisfies the Taylor-Wiles hypothesis and some technical genericity assumptions. For v a fixed place of F above p , we prove that many of the admissible smooth representations of GL 2 (F v) over F p ‾ associated to r ‾ in the corresponding Hecke-eigenspaces of the mod p cohomology have Gelfand–Kirillov dimension [ F v : Q p ]. This builds on and extends the work of Breuil-Herzig-Hu-Morra-Schraen in [2] and Hu-Wang in [12] , giving a unified proof in all cases (r ‾ either semisimple or not at v). [ABSTRACT FROM AUTHOR]

In this article, we prove that every ultragraph Leavitt path algebra is a direct limit of Leavitt path algebras of finite graphs and determine the Gelfand-Kirillov dimension of an ultragraph Leavitt path algebra. We also characterize ultragraph Leavitt path algebras whose simple modules are finitely presented, and show that these algebras have finite Gelfand-Kirillov dimension. Moreover, we construct new classes of simple modules over ultragraph Leavitt path algebras associated with minimal infinite emitters and minimal sinks, which have not appeared in the context of Leavitt path algebras of graphs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Nichols algebra B (W X , r) associated to any non-degenerate involutive solution (X, r) of the Yang-Baxter equation. Infinite examples of finite dimensional Nichols algebras are obtained, including those of dimension nm with m, n ∈ Z ≥ 2 . It turns out that the Nichols algebra B (W X , r) has interesting relations with multinomial expansion. This is a generalization of the work in arXiv:2103.06489, which built a connection between the Nichols algebras of squared dimension and Pascal's triangle. [ABSTRACT FROM AUTHOR]

We study the growth and Gelfand-Kirillov dimension (GK-dimension) of the generalized Weyl algebra (GWA) A = D(σ, a), where D is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for GKdim(A) = GKdim(D) + 1 are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, namely, GKdim(A) is either 3 or ∞ in this case. Our results generalize several existing results in the literature and can be applied to determine the growth, GK-dimension, simplicity and cancellation properties of some GWAs. [ABSTRACT FROM AUTHOR]

We study the Frobenius skew polynomial ring over a polynomial ring in finitely many variables and show that growth function associated to the variables and the Frobenius satisfies a linear recurrence. We introduce the concept of Gelfand-Kirillov base for a finitely generated standard graded algebra over the prime field of characteristic p, where p is prime, and show its existence for the polynomial ring. [ABSTRACT FROM AUTHOR]

We show that every finite GK-dimensional pre-Nichols algebra for braidings of diagonal type with connected diagram of modular, supermodular or unidentified type is a quotient of the distinguished pre-Nichols algebra introduced by the first-named author, up to two exceptions. For these two exceptional cases, we provide a pre-Nichols algebra that substitutes the distinguished one in the sense that it projects onto all finite GK-dimensional pre-Nichols algebras. We build both substitutes as non-trivial central extensions with finite GK-dimension of the corresponding distinguished pre-Nichols algebra. We describe these algebras by generators and relations and give a basis. This work essentially completes the study of eminent pre-Nichols algebras of diagonal type with connected diagram and finite-dimensional Nichols algebra. [ABSTRACT FROM AUTHOR]

Hecke–Kiselman algebras A Θ , over a field 핂 , associated to finite oriented graphs Θ are considered. It has been known that every such algebra is an automaton algebra in the sense of Ufranovskii. In particular, its Gelfand–Kirillov dimension is an integer if it is finite. In this paper, a numerical invariant of the graph Θ that determines the dimension of A Θ is found. Namely, we prove that the Gelfand–Kirillov dimension of A Θ is the sum of the number of cyclic subgraphs of Θ and the number of oriented paths of a special type in the graph, each counted certain specific number of times. [ABSTRACT FROM AUTHOR]

We first offer a fast method for calculating the Gelfand-Kirillov dimension of a finitely presented commutative algebra by investigating certain finite set. Then we establish a Gröbner-Shirshov bases theory for bicommutative algebras, and show that every finitely generated bicommutative algebra has a finite Gröbner-Shirshov basis. As an application, we show that the Gelfand-Kirillov dimension of a finitely generated bicommutative algebra is a nonnegative integer. [ABSTRACT FROM AUTHOR]

Let H be a connected graded Hopf algebra over a field of characteristic zero and K an arbitrary graded Hopf subalgebra of H. We show that there is a family of homogeneous elements of H indexed by a totally order set that satisfy several desirable conditions, which reveal interesting connections between H and K. In particular, the set on non-decreasing products of these elements give a basis for H as a left and right graded K -module. As one of its consequences, we see that H is a graded iterated Hopf Ore extension of K of derivation type provided that H is of finite Gelfand-Kirillov dimension. The main tool of this work is Lyndon words, along the idea developed by Lu, Shen and the second-named author in [24]. [ABSTRACT FROM AUTHOR]

In [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras and which depend on a parameter q and two polynomials f , g. We have shown that this class includes all generalized Heisenberg algebras (as defined in [E. M. F. Curado and M. A. Rego-Monteiro, Multi-parametric deformed Heisenberg algebras: A route to complexity, J. Phys. A: Math. Gen. 34(15) (2001) 3253; R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291, MR 3325233]) as well as generalized down-up algebras (as defined in [G. Benkart and T. Roby, Down-up algebras, J. Algebra 209(1) (1998) 305–344; T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279(1) (2004) 402–421, MR 2078408 (2005f:16051)]), but the parameters of freedom we allow for give rise to many algebras which are in neither one of these two classes. Having classified their finite-dimensional irreducible representations in [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301], in this paper, we turn to their classification by isomorphism, the description of their automorphism groups and the study of their ring-theoretical properties. [ABSTRACT FROM AUTHOR]

We complete the classification of the pointed Hopf algebras with finite Gelfand-Kirillov dimension that are liftings of the Jordan plane over a nilpotent-by-finite group, correcting the statement in [N. Andruskiewitsch, I. Angiono and I. Heckenberger, Liftings of Jordan and super Jordan planes, Proc. Edinb. Math. Soc., II. Ser. 61(3) (2018), 661–672.]. [ABSTRACT FROM AUTHOR]

The Gelfand-Kirillov dimension measures the asymptotic rate of growth of algebras. For every associative dialgebra D , the quotient A D := D / I d (S) , where I d (S) is the ideal of D generated by the set S := { x ⊢ y − x ⊣ y ∣ x , y ∈ D } , is called the associative algebra associated to D . We show that G K d i m (D) ≤ 2 G K d i m (A D). Moreover, we prove that no associative dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2. [ABSTRACT FROM AUTHOR]

Let k be a field and let A be a finitely generated k-algebra. The algebra A is said to be cancellative if whenever B is another k-algebra with the property that A [ x ] ≅ B [ x ] then we necessarily have A ≅ B . An important result of Abhyankar, Eakin, and Heinzer shows that if A is a finitely generated commutative integral domain of Krull dimension one then it is cancellative. We consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand–Kirillov dimension one, and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand–Kirillov dimension one need not be cancellative when the base field has positive characteristic, giving a counterexample to a conjecture of Tang, the fourth-named author, and Zhang. In addition, we prove a skew analogue of the result of Abhyankar–Eakin–Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings. [ABSTRACT FROM AUTHOR]

Let A be a right noetherian algebra over a field k. If the base field extension A ⊗kK remains right noetherian for all extension fields K of k, then A is called stably right noetherian over k. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many ℕ -graded rings of finite global dimension; in particular, we see that a noetherian Artin-Schelter regular algebra must be stably noetherian. In addition, we study more general variations of the stably noetherian property where the field extensions are restricted to those of a certain type, for instance purely transcendental extensions. [ABSTRACT FROM AUTHOR]

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number b n of minimal varieties of given exponent n is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit β = lim n → ∞ b n n exists and can be expressed as the positive solution of an equation a (t) = 0 where a (t) is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank d. It follows from classical results on lacunary power series that the generating function of the sequence b n , n = 1 , 2 , ... , is transcendental. With the same approach we construct examples of free graded semigroups 〈 Y 〉 with the following property. If d n is the number of elements of degree n of 〈 Y 〉 , then the limit δ = lim n → ∞ d n n exists and is transcendental. [ABSTRACT FROM AUTHOR]

The question of whether a noncommutative graded quotient singularity AG is isolated depends on a subtle invariant of the G-action on A, called the pertinency. We prove a partial dichotomy theorem for isolatedness, which applies to a family of noncommutative quotient singularities arising from a graded cyclic action on the (−1)-skew polynomial ring. Our results generalize and extend some results of Bao, He, and the third-named author and results of Gaddis, Kirkman, Moore, and Won. [ABSTRACT FROM AUTHOR]

Let A be a right Ore domain, Z(A) be the center of A and Qr(A) be the right total ring of fractions of A. If K is a field and A is a K-algebra, in this short paper we prove that if A is finitely generated and GKdim(A) < GKdim(Z(A)) + 1, then Z(Qr(A)) ≅= Q(Z(A)). Many examples that illustrate the theorem are included, most of them within the skew PBW extensions. [ABSTRACT FROM AUTHOR]

We determine the Gelfand–Kirillov dimension of a weighted Leavitt path algebra L K (E , w) where K is a field and (E , w) a row-finite weighted graph. Further we show that a finite-dimensional weighted Leavitt path algebra over K is isomorphic to a finite product of matrix rings over K. [ABSTRACT FROM AUTHOR]

We prove that finite GK-dimensional pre-Nichols algebras of super and standard type are quotients of the corresponding distinguished pre-Nichols algebras, except when the braiding matrix is of type super A and the dimension of the braided vector space is three. For these two exceptions we explicitly construct substitutes as braided central extensions of the corresponding pre-Nichols algebras by a polynomial ring in one variable. Via bosonization this gives new examples of finite GK-dimensional Hopf algebras. [ABSTRACT FROM AUTHOR]

We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand–Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated, respectively.

In this note, we compute the Gelfand-Kirillov dimension of cosemisimple Hopf algebras that arise as deformations of a linearly reductive algebraic group. Our work lies in a purely algebraic setting and generalizes results of Goodearl-Zhang (2007), of Banica-Vergnioux (2009), and of D'Andrea-Pinzari- Rossi (2017). [ABSTRACT FROM AUTHOR]

The Gelfand-Kirillov dimension is an invariant which can measure the size of infinite-dimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma modules. In particular, we use it to determine the reducibility of scalar generalized Verma modules associated with maximal parabolic subalgebras in the Hermitian symmetric case. [ABSTRACT FROM AUTHOR]

Tauvel's height formula, which provides a link between the height of a prime ideal and the Gelfand–Kirillov dimension of the corresponding factor algebra, is verified for quantum nilpotent algebras. [ABSTRACT FROM AUTHOR]

We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual) εn! of the Fomin-Kirillov algebras εn; these algebras are connected N-graded and are defined for n ≥ 2. We establish that the algebra εn! is module finite over its cεnter (and thus satisfies a polynomial idεntity), is Noetherian, and has Gelfand-Kirillov dimεnsion ⌊n/2⌋ for each n ≥ 2. We also observe that εn! is not prime for n ≥ 3. By a result of Roos, εn is not Koszul for n ≥ 3, so neither is εn! for n ≥ 3. Nevertheless, we prove that εn! is Artin-Schelter (AS-)regular if and only if n = 2, and that εn! is both AS-Gorεnstein and AS-Cohεn-Macaulay if and only if n = 2, 3. We also show that the depth of εn! is ≤ 1 for each n ≥ 2, conjecture that we have equality, and show that this claim holds for n = 2, 3. Several other directions for further examination of εn! are suggested at the εnd of this article. [ABSTRACT FROM AUTHOR]

In this article, we prove that the Gelfand-Kirillov dimension of the quantized algebra of regular functions on certain homogeneous spaces of types A, C, and D is equal to the dimension of the homogeneous space as a real differentiable manifold. [ABSTRACT FROM AUTHOR]

Abstract We prove a strict relation between the Gelfand–Kirillov (GK) dimension of the relatively free (graded) algebra of a PI-algebra and its (graded) exponent. As a consequence we show a Bahturin–Zaicev type result relating the GK dimension of the relatively free algebra of a graded PI-algebra and the one of its neutral part. We also get that the growth of the relatively free graded algebra of a matrix algebra is maximal when the grading is fine. Finally we compute the graded GK dimension of the matrix algebra with any grading and the graded GK dimension of any verbally prime algebra endowed with an elementary grading. [ABSTRACT FROM AUTHOR]

While it was identified that the growth of any connected Hopf algebras is either a positive integer or infinite, we have yet to determine the Gelfand–Kirillov (GK) dimension of a given connected Hopf algebra. We use the notion of anti-cocommutative elements introduced in Wang, D. G., Zhang, J. J., Zhuang, G. (2013). Coassociative lie algebras. Glasgow Math. J. 55(A):195–215 to analyze the structure of connected Hopf algebras generated by anti-cocommutative elements and compute the GK dimension of said algebras. Additionally, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements. [ABSTRACT FROM AUTHOR]

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand-Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand-Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc. [ABSTRACT FROM AUTHOR]

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms: the first one is between the characteristic Lie algebra χ(sinhu) of the sinh-Gordon equation uxy=sinhu and the non-negative part ℒ(픰픩(2,ℂ))≥0 of the loop algebra of 픰픩(2,ℂ) that corresponds to the Kac-Moody algebra A1(1)χ(sinhu)≅ℒ(픰픩(2,ℂ))≥0=픰픩(2,ℂ)⊗ℂ[t].the second isomorphism is for the Tzitzeica equation uxy = eu + e− 2uχ(eu+e−2u)≅ℒ(픰픩(3,ℂ),μ)≥0=⊕j=0+∞픤j(mod2)⊗tj, where ℒ(픰픩(3,ℂ),μ)=⊕j∈ℤ픤j(mod2)⊗tj is the twisted loop algebra of the simple Lie algebra 픰픩(3,ℂ) that corresponds to the Kac-Moody algebra A2(2).Hence the Lie algebras χ(sinhu) and χ(eu + e− 2u) are slowly linearly growing Lie algebras with average growth rates 32 and 43 respectively. [ABSTRACT FROM AUTHOR]

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra ℬ (V) of a Yetter–Drinfeld module V on which a Lie algebra 𝔤 acts by biderivations. As a special case, we find unrolled versions of the small quantum group. [ABSTRACT FROM AUTHOR]

In this paper, we give a method to compute the Gelfand-Kirillov dimensions of some polynomial-type weight modules. These modules are infinite-dimensional irreducible 픬(n,ℂ)-modules and 픭(2n,ℂ)-modules that appeared in the ℤ-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules. [ABSTRACT FROM AUTHOR]

We classify pointed Hopf algebras with finite Gelfand–Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either a Jordan or a super Jordan plane over a nilpotent-by-finite group. [ABSTRACT FROM AUTHOR]

Let K be a field, and a finitely generated K-algebra with the PBW K-basis . It is shown that if L is a nonzero left ideal of A with GK.dim(A∕L) = d<n ( =  the number of generators of A), then L has the elimination property in the sense that V(U)∩L≠{0} for every subset with , where V(U) = K-span. In terms of the structural properties of A, it is also explored when the condition GK.dim(A∕L)<n may hold for a left ideal L of A. Moreover, from the viewpoint of realizing the elimination property by means of Gröbner bases, it is demonstrated that if A is in the class of binomial skew polynomial rings or in the class of solvable polynomial algebras, then every nonzero left ideal L of A satisfies GK.dim(A∕L)< GK.dimA = n ( =  the number of generators of A), thereby L has the elimination property. [ABSTRACT FROM AUTHOR]

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. As their natural analogues, there are constructions of nil Lie p -algebras over a field of characteristic 2 [40] and arbitrary positive characteristic [52] . In characteristic zero, similar examples of Lie algebras do not exist by a result of Martinez and Zelmanov [32] . The second author constructed analogues of the Grigorchuk and Gupta-Sidki groups in the world of Lie superalgebras of arbitrary characteristic, the virtue of that construction is that the Lie superalgebras have clear monomial bases [41] . That Lie superalgebras have slow polynomial growth and are graded by multidegree in the generators. In particular, a self-similar Lie superalgebra Q is Z 3 -graded by multidegree in 3 generators, its Z 3 -components lie inside an elliptic paraboloid in space, the components are at most one-dimensional, thus, the Z 3 -grading of Q is fine. An analogue of the periodicity is that homogeneous elements of the grading Q = Q 0 ¯ ⊕ Q 1 ¯ are ad-nilpotent. In particular, Q is a nil finely graded Lie superalgebra, which shows that an extension of the mentioned result of Martinez and Zelmanov [32] to the Lie superalgebras of characteristic zero is not valid. But computations with Q are rather technical. In this paper, we construct a similar but simpler and “smaller” example. Namely, we construct a 2-generated fractal Lie superalgebra R over arbitrary field. We find a clear monomial basis of R and, unlike many examples studied before, we find also a clear monomial basis of its associative hull A , the latter has a quadratic growth. The algebras R and A are Z 2 -graded by multidegree in the generators, positions of their Z 2 -components are bounded by pairs of logarithmic curves on plane. The Z 2 -components of R are at most one-dimensional, thus, the Z 2 -grading of R is fine. As an analogue of periodicity, we establish that homogeneous elements of the grading R = R 0 ¯ ⊕ R 1 ¯ are ad-nilpotent. In case of N -graded algebras, a close analogue to being simple is being just infinite. Unlike previous examples of Lie superalgebras, we are able to prove that R is just infinite, but not hereditary just infinite. Our example is close to the smallest possible example, because R has a linear growth with a growth function γ R ( m ) ≈ 3 m , as m → ∞ . Moreover, R is of finite width 4 ( char K ≠ 2 ). In case char K = 2 , we obtain a Lie algebra of width 2 that is not thin. Thus, we have got a more handy analogue of the Grigorchuk and Gupta-Sidki groups. The constructed Lie superalgebra R is of linear growth, of finite width 4, and just infinite. It also shows that an extension of the result of Martinez and Zelmanov [32] to the Lie superalgebras of characteristic zero is not valid. [ABSTRACT FROM AUTHOR]

The Gelfand-Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand-Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489-493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand-Kirillov dimension of an induced module. [ABSTRACT FROM AUTHOR]

We study some properties of the Gelfand-Kirillov dimension in a non-necessarily unital context, in particular, its Morita invariance when the algebras have local units, and its commutativity with direct limits. We then give some applications in the context of graph algebras, which embraces, among some others, path algebras and Cohn and Leavitt path algebras. In particular, we determine the GK-dimension of these algebras in full generality, so extending the main result in A. Alahmadi, H. Alsulami, S. K. Jain and E. Zelmanov, Leavitt Path algebras of finite Gelfand-Kirillov dimension, J. Algebra Appl. 11(6) (2012) 1250225-1250231. [ABSTRACT FROM AUTHOR]

An infinite-dimensional representation π of a real reductive Lie group G can often be thought of as a function space on some manifold X. Although X is not uniquely defined by π, there are 'geometric invariants' of π, first introduced by Roger Howe in the 1970s, related to the geometry of X. These invariants are easy to define but difficult to compute. I will describe some of the invariants, and recent progress toward computing them. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré-Birkhoff-Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand-Kirillov dimension. We will extend the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre-Artin-Zhang-Verevkin theorem. Some examples are included at the end of the paper. [ABSTRACT FROM AUTHOR]

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras. [ABSTRACT FROM AUTHOR]

Primitive cohomology of a Hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra. Among many of its applications, two classifications are presented. Firstly we classify all non-locally PI, pointed Hopf algebra domains of Gelfand–Kirillov dimension two; and secondly we classify all pointed Hopf algebras of rank one. The first classification extends some results of Brown, Goodearl and others in an ongoing project to understand all Hopf algebras of low Gelfand–Kirillov dimension. The second generalizes results of Krop–Radford and Wang–You–Chen which classified Hopf algebras of rank one under extra hypothesis. Properties and algebraic structures of the primitive cohomology are discussed. [ABSTRACT FROM AUTHOR]

Then-dimensional quantum torus 𝒪q((F×)n) is defined as the associativeF-algebra generated byx1,…,xntogether with their inverses satisfying the relationsxixj = qijxjxi, where q = (qij). We show that the modules that are finitely generated over certain commutative sub-algebras ℬ are ℬ-torsion-free and have finite length. We determine the Gelfand–Kirillov dimensions of simple modules in the case whenwhere K.dim stands for the Krull dimension. In this case, ifMis a simple 𝒪q((F×)n)-module, then 𝒢𝒦-dim(M) = 1 orwhere 𝒵(C) stands for the center of an algebraC. We also show that there always exists a simpleF*A-module satisfying the above inequality. [ABSTRACT FROM PUBLISHER]

Leavitt path algebras of an arbitrary graph over a field satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When is a finite graph, satisfying a polynomial identity is shown to be equivalent to the Gelfand-Kirillov dimension of being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph , the Leavitt path algebra has Gelfand-Kirillov dimension zero if and only if has no cycles. Likewise, has Gelfand-Kirillov dimension one if and only if contains at least one cycle, but no cycle in has an exit. [ABSTRACT FROM AUTHOR]